Parametrized homotopic distance
Navnath Daundkar, J. M. García-Calcines
TL;DR
This work introduces and develops the parametrized (fibrewise) homotopic distance $D_B(f,g)$, unifying fibrewise versions of Lusternik–Schnirelman category and topological complexity within a single invariant. It establishes a tight link between $D_B(f,g)$ and the fibrewise sectional category, yields cohomological and dimension-connectivity lower/upper bounds, and analyzes behavior under composition, product, and fibrewise fibrations. The authors study fibrewise $H$-spaces, proving that those with a fibrewise division map satisfy $ ext{TC}_B(X)= ext{cat}_B^{*}(X)$ and providing sharp TC bounds for sphere-bundle examples; they also develop a pointed analogue $D_B^B$ with its own cohomological bounds and show conditions under which the pointed and unpointed invariants coincide. Finally, they compare pointed vs. unpointed distances, giving criteria for when they differ by at most one and when they agree, thereby extending several results in the fibrewise setting such as Calcines–Fibrewise TC.
Abstract
We introduce the concept of parametrized homotopic distance, extending the classical notion of homotopic distance to the fibrewise setting. We establish its correspondence with the fibrewise sectional category of a specific fibrewise fibration and derive cohomological lower bounds and connectivity upper bounds under mild conditions. We also analyze the behavior of parametrized homotopic distance under compositions and products of fibrewise maps, along with its interaction with the triangle inequality. We establish several sufficient conditions for fibrewise $H$-spaces to admit a fibrewise division map and prove that their parametrized topological complexity equals their fibrewise unpointed Lusternik-Schnirelman category, extending Lupton and Scherer's theorem to the fibrewise setting. Additionally, we give sharp estimates for the parametrized topological complexity of a class fibrewise $H$-spaces which arises as sphere bundles with fibre $S^7$. Furthermore, we estimate the parametrized homotopic distance of fibre-preserving, fibrewise maps between fibrewise fibrations, in terms of the parametrized homotopic distance of the induced fibrewise maps between individual fibres, as well as the fibrewise unpointed Lusternik-Schnirelman category of the base space. Finally, we define and study a pointed version of parametrized homotopic distance, establishing cohomological bounds and identifying key conditions for its equivalence with the unpointed version, thus providing a finer classification of fibrewise homotopy invariants.